3.44.26 \(\int \frac {e^{\frac {25 x^4+(x+400 x^3+25 x^4) \log (x)}{25 x+(400+25 x) \log (x)}} (x+75 x^4+(2400 x^3+150 x^4) \log (x)+(16+19200 x^2+2400 x^3+75 x^4) \log ^2(x))}{25 x^2+(800 x+50 x^2) \log (x)+(6400+800 x+25 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ 1+e^{x \left (x^2+\frac {1}{25 \left (16+x+\frac {x}{\log (x)}\right )}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 6.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x)*Log[x]))*(x + 75*x^4 + (2400*x^3 + 150*x
^4)*Log[x] + (16 + 19200*x^2 + 2400*x^3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 + 800*x
+ 25*x^2)*Log[x]^2),x]

[Out]

3*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))*x^2, x] + (16*Defer[
Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/(16 + x)^2, x])/25 + (16*Def
er[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/(x + 16*Log[x] + x*Log[x]
)^2, x])/25 + Defer[Int][(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))*x)/(x +
16*Log[x] + x*Log[x])^2, x]/25 + (4096*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log
[x] + x*Log[x])))/((16 + x)^2*(x + 16*Log[x] + x*Log[x])^2), x])/25 - (512*Defer[Int][E^((25*x^4 + (x + 400*x^
3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/((16 + x)*(x + 16*Log[x] + x*Log[x])^2), x])/25 + (512*De
fer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/((16 + x)^2*(x + 16*Log[
x] + x*Log[x])), x])/25 - (32*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Lo
g[x])))/((16 + x)*(x + 16*Log[x] + x*Log[x])), x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 (x+16 \log (x)+x \log (x))^2} \, dx\\ &=\frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{(x+16 \log (x)+x \log (x))^2} \, dx\\ &=\frac {1}{25} \int \left (\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (16+19200 x^2+2400 x^3+75 x^4\right )}{(16+x)^2}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x \left (256+48 x+x^2\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2}-\frac {32 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(16+x)^2 (x+16 \log (x)+x \log (x))}\right ) \, dx\\ &=\frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (16+19200 x^2+2400 x^3+75 x^4\right )}{(16+x)^2} \, dx+\frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x \left (256+48 x+x^2\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2} \, dx-\frac {32}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(16+x)^2 (x+16 \log (x)+x \log (x))} \, dx\\ &=\frac {1}{25} \int \left (75 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x^2+\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2}\right ) \, dx+\frac {1}{25} \int \left (\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(x+16 \log (x)+x \log (x))^2}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(x+16 \log (x)+x \log (x))^2}+\frac {4096 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2}-\frac {512 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))^2}\right ) \, dx-\frac {32}{25} \int \left (-\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))}\right ) \, dx\\ &=\frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(x+16 \log (x)+x \log (x))^2} \, dx+\frac {16}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2} \, dx+\frac {16}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(x+16 \log (x)+x \log (x))^2} \, dx-\frac {32}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))} \, dx+3 \int \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x^2 \, dx-\frac {512}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))^2} \, dx+\frac {512}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))} \, dx+\frac {4096}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.19, size = 69, normalized size = 2.76 \begin {gather*} e^{x^3+\frac {x}{400+25 x}+\frac {x^4}{x+(16+x) \log (x)}} x^{-\frac {x^2+400 x^4+25 x^5}{25 (16+x) \log (x) (x+(16+x) \log (x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x)*Log[x]))*(x + 75*x^4 + (2400*x^3 +
150*x^4)*Log[x] + (16 + 19200*x^2 + 2400*x^3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 +
800*x + 25*x^2)*Log[x]^2),x]

[Out]

E^(x^3 + x/(400 + 25*x) + x^4/(x + (16 + x)*Log[x]))/x^((x^2 + 400*x^4 + 25*x^5)/(25*(16 + x)*Log[x]*(x + (16
+ x)*Log[x])))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 34, normalized size = 1.36 \begin {gather*} e^{\left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 400 \, x^{3} + x\right )} \log \relax (x)}{25 \, {\left ({\left (x + 16\right )} \log \relax (x) + x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="fricas")

[Out]

e^(1/25*(25*x^4 + (25*x^4 + 400*x^3 + x)*log(x))/((x + 16)*log(x) + x))

________________________________________________________________________________________

giac [B]  time = 0.22, size = 72, normalized size = 2.88 \begin {gather*} e^{\left (\frac {x^{4} \log \relax (x)}{x \log \relax (x) + x + 16 \, \log \relax (x)} + \frac {x^{4}}{x \log \relax (x) + x + 16 \, \log \relax (x)} + \frac {16 \, x^{3} \log \relax (x)}{x \log \relax (x) + x + 16 \, \log \relax (x)} + \frac {x \log \relax (x)}{25 \, {\left (x \log \relax (x) + x + 16 \, \log \relax (x)\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="giac")

[Out]

e^(x^4*log(x)/(x*log(x) + x + 16*log(x)) + x^4/(x*log(x) + x + 16*log(x)) + 16*x^3*log(x)/(x*log(x) + x + 16*l
og(x)) + 1/25*x*log(x)/(x*log(x) + x + 16*log(x)))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 39, normalized size = 1.56

method	result	size
risch	\({\mathrm e}^{\frac {x \left (25 x^{3} \ln \relax (x )+400 x^{2} \ln \relax (x )+25 x^{3}+\ln \relax (x )\right )}{25 x \ln \relax (x )+400 \ln \relax (x )+25 x}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((75*x^4+2400*x^3+19200*x^2+16)*ln(x)^2+(150*x^4+2400*x^3)*ln(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)*ln(x)+2
5*x^4)/((25*x+400)*ln(x)+25*x))/((25*x^2+800*x+6400)*ln(x)^2+(50*x^2+800*x)*ln(x)+25*x^2),x,method=_RETURNVERB
OSE)

[Out]

exp(1/25*x*(25*x^3*ln(x)+400*x^2*ln(x)+25*x^3+ln(x))/(x*ln(x)+16*ln(x)+x))

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 3.61, size = 64, normalized size = 2.56 \begin {gather*} x^{\frac {x^4+16\,x^3}{x+16\,\ln \relax (x)+x\,\ln \relax (x)}+\frac {x}{25\,x+400\,\ln \relax (x)+25\,x\,\ln \relax (x)}}\,{\mathrm {e}}^{\frac {25\,x^4}{25\,x+400\,\ln \relax (x)+25\,x\,\ln \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x)*(x + 400*x^3 + 25*x^4) + 25*x^4)/(25*x + log(x)*(25*x + 400)))*(x + log(x)*(2400*x^3 + 150*x^
4) + log(x)^2*(19200*x^2 + 2400*x^3 + 75*x^4 + 16) + 75*x^4))/(log(x)^2*(800*x + 25*x^2 + 6400) + log(x)*(800*
x + 50*x^2) + 25*x^2),x)

[Out]

x^((16*x^3 + x^4)/(x + 16*log(x) + x*log(x)) + x/(25*x + 400*log(x) + 25*x*log(x)))*exp((25*x^4)/(25*x + 400*l
og(x) + 25*x*log(x)))

________________________________________________________________________________________

sympy [A]  time = 0.70, size = 32, normalized size = 1.28 \begin {gather*} e^{\frac {25 x^{4} + \left (25 x^{4} + 400 x^{3} + x\right ) \log {\relax (x )}}{25 x + \left (25 x + 400\right ) \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x**4+2400*x**3+19200*x**2+16)*ln(x)**2+(150*x**4+2400*x**3)*ln(x)+75*x**4+x)*exp(((25*x**4+400*
x**3+x)*ln(x)+25*x**4)/((25*x+400)*ln(x)+25*x))/((25*x**2+800*x+6400)*ln(x)**2+(50*x**2+800*x)*ln(x)+25*x**2),
x)

[Out]

exp((25*x**4 + (25*x**4 + 400*x**3 + x)*log(x))/(25*x + (25*x + 400)*log(x)))

________________________________________________________________________________________